Abstract:
We consider the problem of enstrophy dissipation for twodimensional
incompressible flows at high Reynolds number. We discuss two notions of
enstrophy defects associated to approximate solution sequences to the Euler
equations obtained by mollification and by vanishing viscosity. These notions
were originally introduced by G. Eyink in order to reconcile the
KraichnanBatchelor theory of 2D turbulence with properties of weak solutions to
2D Euler. We show that if the initial enstrophy is finite, the viscous enstrophy
defect is welldefined. If the initial vorticity belongs to certain logarithmic
refinements of L^2, then an exact transport equation holds for the corresponding
enstrophy density, because of cancellation properties of the BiotSavart kernel.
Moreover, the transport defect also exists. For rougher data in a Besov space,
for which the initial enstrophy is infinite, the enstrophy defects depend upon
the approximating sequence and we produce examples of solutions where true
dissipation occurs. This is joint work with Helena and Milton Lopes.
[LECTURE SLIDES]
